### Home > APCALC > Chapter 3 > Lesson 3.2.2 > Problem 3-64

Using the *definition of the derivative* as a limit, show that the derivative of

**The Derivative**

The slope of a line tangent to at any point* ***derivative **of at

*. The standard form of this type of limit is:*

If the slope (or instantaneous rate of change) at a particular -value is desired, such as at

*. This slope can be found by replacing*

*with*

*.*

This is one form of the 'definition of the derivative' (informally known as Hana's Method).

In order to evaluate this limit, we need to find an Algebraic way to cancel out the in the denominator.

Find a common denominator in the numerator, expand and combine like terms:

Factor the numerator, then cancel out the :

Since there is no longer and in the denominator, you can evaluate the limit as

*:*